16 Mr. W. Barlow on Crystal Symmetry. 



Now if the mutual inclination of the two axes is infinit- 

 esimal, the length AB on the sphere which is intercepted by 

 the lines drawn parallel to them will be infinitesimal as com- 

 pared with the dimensions of the sphere ; consequently the 



angle r , and therefore also the angle 7, will be infinitesimal. 



But by proposition 8 the latter is impossible, therefore so 

 also is an infinitesimal value for the mutual inclination of the 

 two axes. 



Proposition 10. — Among identical axes present in a homo- 

 geneous structure, parallel ones are always found. 



Proof. It has been shown by proposition 6 that either 

 parallel axes, or such as have their directions inclined to 

 one another at infinitesimal angles, are present, and since 

 by proposition 9 the latter are impossible, the existence of 

 the former is established. 



Proposition 11. — The minimum rotation-angle of an axis 

 of coincidence of a homogeneous structure must be an aliquot 



part of 360° ; i. e. if it be designated — , n must be an 

 integer. 



Proof. The existence of an axis, whether screw-spiral or 

 rotational, whose minimum coincidence-movement produces 

 rotation through an angle a, involves the possibility of co- 

 incidence-movements whose rotation components have angles 

 2a, 3a, &c. ; and if a coincidence-movement can be effected in 

 one direction, it can also be effected in the opposite direction. 

 This is evident when it is considered that the aspect of the 

 system from any fixed point, and its relations as a whole to 

 fixed directions, remain altogether unchanged by a coinci- 

 dence-mo vement, and that an opposite movement can always 

 be regarded as a return to a former position. 



Let pa be the angle of the series 2a } 3«, &c, which is not 

 less than 360° and nearest to this value, p being an integer. 



Then any other value for pa than exactly 360° would give 

 a possible movement with a rotation-component pa— 360° < a. 

 As this is contrary to hypothesis pa must =360°*". 



Proposition 12. — The whole number n which is the rotation- 

 coefficient of an axis I being the minimum rotation-com- 

 ponent) cannot be greater than 6. 



Proof. Among the identical parallel axes which, if a 

 homogeneous structure contains axes, must, according to pro- 

 position 10, be present, let A, A' be two whose distance apart 

 a is as small as that separating any of such axes. Then by 

 » Gadolin Memoire &c. p. 8 (translation, p. 9). 



